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User blog:Edwin Shade 2/Calculus Derived Numbers
PsiCubed2's most recent blog post got me thinking about an aspect of googology that isn't really explored often - googology in calculus. Now, it's been a while since I've done any sort of real calculus beyond simple integration or differentiation of polynomials, but, there are some things I began to wonder about reading Psi's blog. MANY things actually. I want to analyze the properties of infinite sums based off of googological functions. However, I currently lack the know how, so in the meantime here are just the names I've given these sums and their definition. Note that Σma=m,b=n,c=o,...E stands for "the sum of the expression E m-times such that on the first summation, variable a = m, variable b = n, variable c = o, etc...; on the second summation, variable a = m + 1, varible b = n + 1, variable c = o + 1, etc...; on the third summation, variable a = m + 2, varible b = n + 2, variable c = o + 2, etc...;; and so forth until m-summations occur". It may sound complicated but in the examples I give it's quite easy to figure out this is just the normal summation operator adjusted for plaintext. Precursor:Fundamental Sequences The fundamental sequences for the FGH are as follows: ωαnFS = ωαnFS if α is a limit ordinal = ωα - 1n if α is a successor ordinal -fsum Regiment Dufsum: Σ∞k=11/(f2(k)) Trifsum: Σ∞k=11/(f3(k)) Quadrifsum: Σ∞k=11/(f4(k)) Quintifsum: Σ∞k=11/(f5(k)) Sexafsum: Σ∞k=11/(f6(k)) Septafsum: Σ∞k=11/(f7(k)) Octafsum: Σ∞k=11/(f8(k)) Nonifsum: Σ∞k=11/(f9(k)) Decifsum: Σ∞k=11/(f10(k)) Centifsum: Σ∞k=11/(f100(k)) Millifsum: Σ∞k=11/(f1000(k)) Myriafsum: Σ∞k=11/(f10000(k)) Myriafsum: Σ∞k=11/(f10000(k)) Lakhfsum: Σ∞k=11/(f100000(k)) Megafsum: Σ∞k=11/(f1000000(k)) Crorefsum: Σ∞k=11/(f10000000(k)) Okufsum: Σ∞k=11/(f100000000(k)) Gigafsum: Σ∞k=11/(f1000000000(k)) Dialogufsum: Σ∞k=11/(f1010(k)) Omefsum: Σ∞k=11/(fω(k)) Unomefsum: Σ∞k=11/(fω+1(k)) Bimefsum: Σ∞k=11/(fω+2(k)) Trimefsum: Σ∞k=11/(fω+3(k)) Quadrimefsum: Σ∞k=11/(fω+4(k)) Quintimefsum: Σ∞k=11/(fω+5(k)) Sexamefsum: Σ∞k=11/(fω+6(k)) Septamefsum: Σ∞k=11/(fω+7(k)) Octimefsum: Σ∞k=11/(fω+8(k)) Nonimefsum: Σ∞k=11/(fω+9(k)) Dumefsum: Σ∞k=11/(fω2(k)) Tremefsum: Σ∞k=11/(fω3(k)) Tetramefsum: Σ∞k=11/(fω4(k)) Pentamefsum: Σ∞k=11/(fω5(k)) Hexamefsum: Σ∞k=11/(fω6(k)) Heptamefsum: Σ∞k=11/(fω7(k)) Octomefsum: Σ∞k=11/(fω8(k)) Enneamefsum: Σ∞k=11/(fω9(k)) Dekamefsum: Σ∞k=11/(fω10(k)) Monodekamefsum: Σ∞k=11/(fω11(k)) Hectomefsum: Σ∞k=11/(fω100(k)) Kilomefsum: Σ∞k=11/(fω1000(k)) Myriamefsum: Σ∞k=11/(fω10000(k)) Lakhmefsum: Σ∞k=11/(fω100000(k)) Megamefsum: Σ∞k=11/(fω1000000(k)) Croremefsum: Σ∞k=11/(fω10000000(k)) Okumefsum: Σ∞k=11/(fω100000000(k)) Gigamefsum: Σ∞k=11/(fω1000000000(k)) Dialogumefsum: Σ∞k=11/(fω1010(k)) Inbioufsum: Σ∞k=11/(fω2(k)) Intrioufsum: Σ∞k=11/(fω3(k)) Inquadrioufsum: Σ∞k=11/(fω4(k)) Inquintioufsum: Σ∞k=11/(fω5(k)) Insexoufsum: Σ∞k=11/(fω6(k)) Inseptoufsum: Σ∞k=11/(fω7(k)) Inoctoufsum: Σ∞k=11/(fω8(k)) Innonoufsum: Σ∞k=11/(fω9(k)) Indecoufsum: Σ∞k=11/(fω10(k)) Inomeoufsum: Σ∞k=11/(fωω(k)) Inunmeoufsum: Σ∞k=11/(fωω+1(k)) Inbimeoufsum: Σ∞k=11/(fωω+2(k)) Intrimeoufsum: Σ∞k=11/(fωω+3(k)) Inquadrimeoufsum: Σ∞k=11/(fωω+4(k)) Inquintimeoufsum: Σ∞k=11/(fωω+5(k)) Insexameoufsum: Σ∞k=11/(fωω+6(k)) Inseptameoufsum: Σ∞k=11/(fωω+7(k)) Inoctimeoufsum: Σ∞k=11/(fωω+8(k)) Innonimeoufsum: Σ∞k=11/(fωω+9(k)) Indecimeoufsum: Σ∞k=11/(fωω+10(k)) Indumeoufsum: Σ∞k=11/(fωω2(k)) Intremeoufsum: Σ∞k=11/(fωω3(k)) Intetrameoufsum: Σ∞k=11/(fωω4(k)) Inpentameoufsum: Σ∞k=11/(fωω5(k)) Inhexameoufsum: Σ∞k=11/(fωω6(k)) Inheptameoufsum: Σ∞k=11/(fωω7(k)) Inoctomeoufsum: Σ∞k=11/(fωω8(k)) Inenneameoufsum: Σ∞k=11/(fωω9(k)) Indekameoufsum: Σ∞k=11/(fωω10(k)) Ininbiououfsum: Σ∞k=11/(fωω2(k)) Inintrououfsum: Σ∞k=11/(fωω3(k)) Ininquadrououfsum: Σ∞k=11/(fωω4(k)) Ininquintououfsum: Σ∞k=11/(fωω5(k)) Ininsexououfsum: Σ∞k=11/(fωω6(k)) Ininseptououfsum: Σ∞k=11/(fωω7(k)) Ininoctououfsum: Σ∞k=11/(fωω8(k)) Ininnonououfsum: Σ∞k=11/(fωω9(k)) Inindecououfsum: Σ∞k=11/(fωω10(k)) Ininomeououfsum: Σ∞k=11/(fωωω(k)) Inininomeouououfsum: Σ∞k=11/(fωωωω(k)) Ininininomeououououfsum: Σ∞k=11/(fωωωωω(k)) Inininininomeouououououfsum: Σ∞k=11/(fωωωωωω(k)) Ininininininomeououououououfsum: Σ∞k=11/(fωωωωωωω(k)) Epsilfsum: Σ∞k=11/(fε0(k)) Epunsilfsum: Σ∞k=11/(fε1(k)) Epbisilfsum: Σ∞k=11/(fε2(k)) Eptrisilfsum: Σ∞k=11/(fε3(k)) Epquadrisilfsum: Σ∞k=11/(fε4(k)) Epquintisilfsum: Σ∞k=11/(fε5(k)) Epsexasilfsum: Σ∞k=11/(fε6(k)) Epseptasilfsum: Σ∞k=11/(fε7(k)) Epoctisilfsum: Σ∞k=11/(fε8(k)) Epnonisilfsum: Σ∞k=11/(fε9(k)) Epdecisilfsum: Σ∞k=11/(fε10(k)) Epepsilsilfsum: Σ∞k=11/(fεε0(k)) Epepepsilsilsilfsum: Σ∞k=11/(fεεε0(k)) Epepepepsilsilsilsilfsum: Σ∞k=11/(fεεεε0(k)) Epepepepepsilsilsilsilsilfsum: Σ∞k=11/(fεεεεε0(k)) Epepepepepepsilsilsilsilsilsilfsum: Σ∞k=11/(fεεεεεε0(k)) Zetafsum: Σ∞k=11/(fζ0(k)) Zeepsiltafsum: Σ∞k=11/(fζε0(k)) Zezetatafsum: Σ∞k=11/(fζζ0(k)) Zezezetatatafsum: Σ∞k=11/(fζζζ0(k)) Zezezezetatatatafsum: Σ∞k=11/(fζζζζ0(k)) Zezezezezetatatatatafsum: Σ∞k=11/(fζζζζζ0(k)) Zezezezezezetatatatatatafsum: Σ∞k=11/(fζζζζζζ0(k)) Zezezezezezezetatatatatatatafsum: Σ∞k=11/(fζζζζζζζ0(k)) Etlafsum: Σ∞k=11/(f��0(k)) Observations *Most (if not all) of these numbers are between one and two. For Googology, this is pretty "bad". For the sake of naming real numbers which up until now may have never been individually thought of it before however, this is good. It is also an exercise in naming conventions. Tasks *Put numbers in the regiments in order. *Make at least one interesting point about them. *Actually finish the regiments first! Category:Blog posts